Question

find the length of side x in simplest radical form with a rational denominator.
(there is a right triangle with angles 60°, 30°, and 90°, one leg is 4, the other leg is x)
answer attempt 1 out of 2

Explanation:

Step1: Identify the triangle type

This is a right - triangle with angles $30^{\circ}$, $60^{\circ}$ and $90^{\circ}$. In a $30 - 60 - 90$ triangle, the sides are in the ratio $1:\sqrt{3}:2$, where the side opposite $30^{\circ}$ is the shortest side, the side opposite $60^{\circ}$ is $\sqrt{3}$ times the shortest side, and the hypotenuse is twice the shortest side.

Step2: Determine the sides

We know that the side opposite the $30^{\circ}$ angle is $4$? Wait, no. Wait, in the right - triangle, the right angle is between the side of length $4$ and the side of length $x$. The angle of $30^{\circ}$ is opposite the side of length $4$, and the angle of $60^{\circ}$ is opposite the side of length $x$.
In a $30 - 60 - 90$ triangle, if the side opposite $30^{\circ}$ is $a$, the side opposite $60^{\circ}$ is $a\sqrt{3}$, and the hypotenuse is $2a$.
Here, the side opposite $30^{\circ}$ is $4$, so the side opposite $60^{\circ}$ (which is $x$) is $4\times\sqrt{3}=4\sqrt{3}$.
We can also use trigonometric ratios. We know that $\tan(60^{\circ})=\frac{x}{4}$. Since $\tan(60^{\circ})=\sqrt{3}$, we have $\sqrt{3}=\frac{x}{4}$.

Step3: Solve for $x$

Multiply both sides of the equation $\sqrt{3}=\frac{x}{4}$ by $4$. We get $x = 4\sqrt{3}$.

Answer:

$4\sqrt{3}$